
[Sponsors] 
July 27, 2000, 05:42 
magnitudes of eigenvalues VS stiffness

#1 
Guest
Posts: n/a

Dear sir, I read from one numerical fluid book that "The condition and degree of stiffness of the system can be related to the relative magnitudes of the eigenvalues. When the eigenvalues differ greatly in magnitudes, convergence to a steadystate solution is usually slow. This occurs because greatly varying signal speeds appear in the equations and the traditional solution schemes attempt to honor all of them.". Actually I know how to find out the eigenvalues of the system, but I never know the physical meaning of the eigenvalues. Could anyone who know please explain this concept to me? I know that they can be used to find out the solution, but I don't know the physical relation between them and the solution. In the quoted sentences, the author say about signal speeds, what does signal mean? Every flow property? Please explain to me. Thank you very much.
Best regards. Atit Koonsrisuk 

July 28, 2000, 12:14 
Re: magnitudes of eigenvalues VS stiffness

#2 
Guest
Posts: n/a

I am no mathematician but I think I can give you some idea about the relationship between the magnitudes of the eigenvalues and stiffness. If you have a system of coupled partial differential equations of the form, U_t + A U_x = 0 where U is a vector (eg. p, u, T for a onedimensional flow) and A is not a diagonal matrix.
Then A can be diagonalized (under some constraints) by a similarity transformation D = T^1 A T, where T is the transformation matrix and D is the resulting diagonal matrix. Now actually if D is chosen to the eigenvalues of A and T is chosen as the matrix made up of the eigenvectors of A then the similarity transformation works. Applying this transformation the system becomes decoupled with each individual equation being of the form w_t + d w_x = 0; this is like a simple plane wave equation with d as the wave speed. w is called a characteristic variable. So now the time step chosen is a function of 'd' and hence if the d's are quite different in magnitude one has to use the highest one to get the solution thus making the system stiff!!! chidu... 

Thread Tools  
Display Modes  


Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
FSI Mesh stiffness  Andrew.Coughtrie  CFX  3  July 29, 2011 03:50 
Mesh stiffness in 1 way FSI  vmlxb6  CFX  4  February 15, 2011 17:17 
deforming mesh issue : I don't get how stiffness works  bennn  CFX  9  October 12, 2010 15:09 
Efficient Conversion of Local Stiffness Matrix to Global Stiffness  nlane  Main CFD Forum  2  September 14, 2010 18:26 
eigenvalues in propagation problems  Atit Koonsrisuk  Main CFD Forum  5  February 21, 2001 13:34 